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The fractal finite element method for unbounded problems
Author(s) -
Leung A. Y. T.,
Dai H.,
Fok S. L.,
Su R. K. L.
Publication year - 2004
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1097
Subject(s) - finite element method , mathematics , similarity (geometry) , fractal , series (stratigraphy) , stress intensity factor , matrix similarity , mathematical analysis , transformation (genetics) , matrix (chemical analysis) , geometry , stress field , computer science , partial differential equation , structural engineering , image (mathematics) , paleontology , biochemistry , chemistry , materials science , composite material , artificial intelligence , gene , engineering , biology
The fractal finite element method, previously developed for stress intensity factor calculation for crack problems in fracture mechanics, is extended to analyse some unbounded problems in half space. The concepts of geometrical similarity and two‐level finite element mesh are applied to generate an infinite number of self‐similar layers in the far field with a similarity ratio bigger than one; that is, one layer is bigger than the next in size but of the same shape. Only conventional finite elements are used and no new elements are generated. The global interpolating functions in the form of a truncated infinite series are employed to transform the infinite number of nodal variables to a small number of unknown coefficients associated with the global interpolating functions. Taking the advantage of geometrical similarity, transformation for one layer is enough because the relevant entries of the transformed matrix after assembling all layers are infinite geometric series of the similarity ratio and can be summed analytically. Accurate nodal displacements are obtained as shown in the numerical examples. Copyright © 2004 John Wiley & Sons, Ltd.